QCD sum rules for 70-plet baryons

QCD sum rules for the 70-plet baryons are proposed, using unitary pattern of these rules. As it was previously shown, all correlators of QCD sum rules for the 56-plet depend only on two generalized functions of the F and D types, characteristic for the unitary symmetry. The same is shown to be valid for the 70-plet baryons.


Introduction
Magnetic moments of the octet 1/2 − baryons attracted some attention recently in quark models as well as in the QCD sum rule formalism.In the quark model these baryons are assigned usually to the 70-plet representation of SU (6), in contrast to the 1/2 + baryons of the 56-plet (see [1,2] and the references therein).
In a series of papers we have shown that QCD sum rules for couplings and magnetic moments of the octet 1/2 + baryons reproduce unitary symmetry results in terms of F and D quantities ( [5] and references therein).In fact, any QCD sum rule for magnetic moments of the octet baryons exhibits the unitary structure.We take a little old-fashioned QCD sum rule [6] as a simple example.If the correlator is T being time ordering operator, γ means external electromagnetic field while η B is interpolating current with the quantum numbers of the B baryon, then, say, for Σ 0 it could be written (we put to shorten formulae m s = 0 disregarding differences between vacuum expectation values of various flavors) as: Here M 2 is the Borel parameter, the vev's are a = −(2π) 2 MeV and normalization point μ=500 MeV.In the limit of degenerated masses of quarks and vacuum expectation values (vev) it reduces to the relation [7]: Note that this relation goes to the NRQM formula with F = 2/3 and D = 1 and e q → μ q , and from the other side it reproduces the unitary formula by putting values of e q 's for all octet baryons but Λ.
The Λ quantities are reproduced easily with our relation [8].

NRQM and quark-diquark model for 70-plet
With the 70-plet octet baryons we loose this correspondence.QCD sum rules remain essentially the same [9,10] (only with γ 5 inside, to change baryon parity) while NRQM gives μ 70 [1] and references therein).We propose a way to arrive at the desired QCD sum rules in accord with the NRQM limit [11] with the help of quark-diquark model which has led previously to Eq.( 3).Let us take quark wave functions of the 70-plet baryon [1] (and we take Σ 0 ) as (subindices 1,2 mean ↑,↓, correspondingly): (ψ (0 + ) and ψ " (0 + ) are spatial wave functions [1]).To obtain Λ wave functions, and also to control the formulae, we use relations with the exchange of quarks: We follow previous reasoning of our quark-diquark model assuming that photon distinguishes between the diquark of (quasi)similar quarks (qq) and single quark Q [7].That is, we introduce 4 matrix elements of the new operator ωq : For the 56-plet one obtains for Σ 0 (ud, s) with the modified magnetic moment operator e q ωq σ q 3 (disregarding ωq one just returns to NRQM):  But now, with F = 2/3 and D = 1, we return to the 70-plet NRQM result μ Σ 0 = 1/3(μ u + μ d + μ s ), while putting values of e q 's we arrive at magnetic moments of 70-plet baryons in terms of F and D constants [11]: We also note that the transition magnetic moment is equal to zero for the 70-plet, Λ * | μ70 |Σ * 0 =0.It can be proved also with the relation in ref. [8].

QCD sum rules for 70-plet magnetic moments
We take it as a base for the construction of QCD sum rules for the magnetic moments (here for 1/2 + ), ).These formulae yield correct NRQM limit and maintain F, D pattern of the QCD sum rules.We proceed with these QCD sum rules in more detail, and take as an example the QCD sum rules for the baryon magnetic moments [6].The Eq.( 2) according to our formulae Eq.( 7) transforms itself for the 70-plet into: It is seen that sum rules (2) and ( 9) differ drastically.Model predictions for both sum rules and also some results for 1/2 − baryons are given in Table 1.

Conclusions
As a result we have shown the way to construct QCD sum rules for the magnetic moments of the 70plet octet baryons which have the correct NRQM limit.Octet baryons in the 70-plet are analyzed in a way similar to those of the 56-plet.Magnetic moments are written in terms of the D and F quantities characteristic for octet coupling.The main formulae for the magnetic moments are written in such a way as to obtain the NRQM results as well as unitary symmetry ones.In NRQM in the 70-plet octet Tansition magnetic moment is shown to be zero.Borel QCD sum rules are constructed for the magnetic moments of the 70-plet octet.Comparison with some other results in the field are made.
I am grateful to T.M. Aliev for partecipating in part of this work, and to the Workshop Organizers for the warm hospitality.